ENA, Section 1.4, Mentioned, A

Percentage Accurate: 53.1% → 100.0%

Time: 7.3s

Alternatives: 7

Speedup: 9.5×

Specification

\[-0.01 \leq x \land x \leq 0.01\]

Math

\[\begin{array}{l} \\ 1 - \cos x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (cos x)))

C

double code(double x) {
	return 1.0 - cos(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - cos(x)
end function

Java

public static double code(double x) {
	return 1.0 - Math.cos(x);
}

Python

def code(x):
	return 1.0 - math.cos(x)

Julia

function code(x)
	return Float64(1.0 - cos(x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - cos(x);
end

Wolfram

code[x_] := N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \cos x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (cos x)))

C

double code(double x) {
	return 1.0 - cos(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - cos(x)
end function

Java

public static double code(double x) {
	return 1.0 - Math.cos(x);
}

Python

def code(x):
	return 1.0 - math.cos(x)

Julia

function code(x)
	return Float64(1.0 - cos(x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - cos(x);
end

Wolfram

code[x_] := N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\sin x}^{2}}{\cos x - -1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow (sin x) 2.0) (- (cos x) -1.0)))

C

double code(double x) {
	return pow(sin(x), 2.0) / (cos(x) - -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) ** 2.0d0) / (cos(x) - (-1.0d0))
end function

Java

public static double code(double x) {
	return Math.pow(Math.sin(x), 2.0) / (Math.cos(x) - -1.0);
}

Python

def code(x):
	return math.pow(math.sin(x), 2.0) / (math.cos(x) - -1.0)

Julia

function code(x)
	return Float64((sin(x) ^ 2.0) / Float64(cos(x) - -1.0))
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) ^ 2.0) / (cos(x) - -1.0);
end

Wolfram

code[x_] := N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[1 - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{1 - \cos x} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\cos x\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos x\right)\right) + 1} \]

   4. flip-+ N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\cos x\right)\right) \cdot \left(\mathsf{neg}\left(\cos x\right)\right) - 1 \cdot 1}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1}} \]

   5. sqr-neg-rev N/A

      \[\leadsto \frac{\color{blue}{\cos x \cdot \cos x} - 1 \cdot 1}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   6. lift-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos x} \cdot \cos x - 1 \cdot 1}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{\cos x \cdot \color{blue}{\cos x} - 1 \cdot 1}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\cos x \cdot \cos x - \color{blue}{1}}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   9. sub-1-cos N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sin x \cdot \sin x\right)}}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   10. 1-sub-cos N/A

       \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(1 - \cos x \cdot \cos x\right)}\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{1 \cdot 1} - \cos x \cdot \cos x\right)\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   12. lift-cos.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(1 \cdot 1 - \color{blue}{\cos x} \cdot \cos x\right)\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   13. lift-cos.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(1 \cdot 1 - \cos x \cdot \color{blue}{\cos x}\right)\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   14. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 \cdot 1 - \cos x \cdot \cos x\right)\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-{\sin x}^{2}}{\left(-\cos x\right) - 1}} \]

5. Final simplification 100.0%

   \[\leadsto \frac{{\sin x}^{2}}{\cos x - -1} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   x)
  x))

C

double code(double x) {
	return (fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5) * x) * x;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5) * x) * x)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)} \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x\right) \cdot x} \]
6. Add Preprocessing

Alternative 3: 99.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right) \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma -0.041666666666666664 (* x x) 0.5) (* x x)))

C

double code(double x) {
	return fma(-0.041666666666666664, (x * x), 0.5) * (x * x);
}

Julia

function code(x)
	return Float64(fma(-0.041666666666666664, Float64(x * x), 0.5) * Float64(x * x))
end

Wolfram

code[x_] := N[(N[(-0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[1 - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{1 - \cos x} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\cos x\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos x\right)\right) + 1} \]

   4. flip-+ N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\cos x\right)\right) \cdot \left(\mathsf{neg}\left(\cos x\right)\right) - 1 \cdot 1}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1}} \]

   5. sqr-neg-rev N/A

      \[\leadsto \frac{\color{blue}{\cos x \cdot \cos x} - 1 \cdot 1}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   6. lift-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos x} \cdot \cos x - 1 \cdot 1}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{\cos x \cdot \color{blue}{\cos x} - 1 \cdot 1}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\cos x \cdot \cos x - \color{blue}{1}}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   9. sub-1-cos N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sin x \cdot \sin x\right)}}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   10. 1-sub-cos N/A

       \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(1 - \cos x \cdot \cos x\right)}\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{1 \cdot 1} - \cos x \cdot \cos x\right)\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   12. lift-cos.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(1 \cdot 1 - \color{blue}{\cos x} \cdot \cos x\right)\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   13. lift-cos.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(1 \cdot 1 - \cos x \cdot \color{blue}{\cos x}\right)\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1} \]

   14. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 \cdot 1 - \cos x \cdot \cos x\right)\right)}{\left(\mathsf{neg}\left(\cos x\right)\right) - 1}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-{\sin x}^{2}}{\left(-\cos x\right) - 1}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}\right)} \cdot {x}^{2} \]

   3. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)} \cdot {x}^{2} + \frac{1}{2}\right) \cdot {x}^{2} \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24} \cdot {x}^{2}\right)\right)} + \frac{1}{2}\right) \cdot {x}^{2} \]

   5. neg-sub0 N/A

      \[\leadsto \left(\color{blue}{\left(0 - \frac{1}{24} \cdot {x}^{2}\right)} + \frac{1}{2}\right) \cdot {x}^{2} \]

   6. associate-+l- N/A

      \[\leadsto \color{blue}{\left(0 - \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot {x}^{2} \]

   7. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)} \cdot {x}^{2} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right) \cdot {x}^{2}} \]

   9. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot {x}^{2} \]

   10. associate-+l- N/A

       \[\leadsto \color{blue}{\left(\left(0 - \frac{1}{24} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \cdot {x}^{2} \]

   11. neg-sub0 N/A

       \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24} \cdot {x}^{2}\right)\right)} + \frac{1}{2}\right) \cdot {x}^{2} \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right) \cdot {x}^{2}} + \frac{1}{2}\right) \cdot {x}^{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\color{blue}{\frac{-1}{24}} \cdot {x}^{2} + \frac{1}{2}\right) \cdot {x}^{2} \]

   14. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{24}, {x}^{2}, \frac{1}{2}\right)} \cdot {x}^{2} \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot {x}^{2} \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot {x}^{2} \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \frac{1}{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   18. lower-*.f64 100.0

       \[\leadsto \mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

7. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right) \cdot \left(x \cdot x\right)} \]
8. Add Preprocessing

Alternative 4: 99.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma -0.041666666666666664 (* x x) 0.5) x) x))

C

double code(double x) {
	return (fma(-0.041666666666666664, (x * x), 0.5) * x) * x;
}

Julia

function code(x)
	return Float64(Float64(fma(-0.041666666666666664, Float64(x * x), 0.5) * x) * x)
end

Wolfram

code[x_] := N[(N[(N[(-0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}\right)} \cdot x\right) \cdot x \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)} \cdot {x}^{2} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24} \cdot {x}^{2}\right)\right)} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   7. neg-sub0 N/A

      \[\leadsto \left(\left(\color{blue}{\left(0 - \frac{1}{24} \cdot {x}^{2}\right)} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   8. associate-+l- N/A

      \[\leadsto \left(\color{blue}{\left(0 - \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot x\right) \cdot x \]

   9. neg-sub0 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)} \cdot x\right) \cdot x \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right) \cdot x\right) \cdot x} \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right) \cdot x\right)} \cdot x \]

   12. neg-sub0 N/A

       \[\leadsto \left(\color{blue}{\left(0 - \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot x\right) \cdot x \]

   13. associate-+l- N/A

       \[\leadsto \left(\color{blue}{\left(\left(0 - \frac{1}{24} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \cdot x\right) \cdot x \]

   14. neg-sub0 N/A

       \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24} \cdot {x}^{2}\right)\right)} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   15. distribute-lft-neg-in N/A

       \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right) \cdot {x}^{2}} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   16. metadata-eval N/A

       \[\leadsto \left(\left(\color{blue}{\frac{-1}{24}} \cdot {x}^{2} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   17. lower-fma.f64 N/A

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{24}, {x}^{2}, \frac{1}{2}\right)} \cdot x\right) \cdot x \]

   18. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot x\right) \cdot x \]

   19. lower-*.f64 100.0

       \[\leadsto \left(\mathsf{fma}\left(-0.041666666666666664, \color{blue}{x \cdot x}, 0.5\right) \cdot x\right) \cdot x \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x} \]
6. Add Preprocessing

Alternative 5: 99.5% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 0.5 x) x))

C

double code(double x) {
	return (0.5 * x) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * x) * x
end function

Java

public static double code(double x) {
	return (0.5 * x) * x;
}

Python

def code(x):
	return (0.5 * x) * x

Julia

function code(x)
	return Float64(Float64(0.5 * x) * x)
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * x) * x;
end

Wolfram

code[x_] := N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}\right)} \cdot x\right) \cdot x \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)} \cdot {x}^{2} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24} \cdot {x}^{2}\right)\right)} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   7. neg-sub0 N/A

      \[\leadsto \left(\left(\color{blue}{\left(0 - \frac{1}{24} \cdot {x}^{2}\right)} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   8. associate-+l- N/A

      \[\leadsto \left(\color{blue}{\left(0 - \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot x\right) \cdot x \]

   9. neg-sub0 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)} \cdot x\right) \cdot x \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right) \cdot x\right) \cdot x} \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right) \cdot x\right)} \cdot x \]

   12. neg-sub0 N/A

       \[\leadsto \left(\color{blue}{\left(0 - \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot x\right) \cdot x \]

   13. associate-+l- N/A

       \[\leadsto \left(\color{blue}{\left(\left(0 - \frac{1}{24} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \cdot x\right) \cdot x \]

   14. neg-sub0 N/A

       \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24} \cdot {x}^{2}\right)\right)} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   15. distribute-lft-neg-in N/A

       \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right) \cdot {x}^{2}} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   16. metadata-eval N/A

       \[\leadsto \left(\left(\color{blue}{\frac{-1}{24}} \cdot {x}^{2} + \frac{1}{2}\right) \cdot x\right) \cdot x \]

   17. lower-fma.f64 N/A

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{24}, {x}^{2}, \frac{1}{2}\right)} \cdot x\right) \cdot x \]

   18. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot x\right) \cdot x \]

   19. lower-*.f64 100.0

       \[\leadsto \left(\mathsf{fma}\left(-0.041666666666666664, \color{blue}{x \cdot x}, 0.5\right) \cdot x\right) \cdot x \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 99.6%

   \[\leadsto \left(0.5 \cdot x\right) \cdot x \]
9. Add Preprocessing

Alternative 6: 99.5% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) 0.5))

C

double code(double x) {
	return (x * x) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * 0.5d0
end function

Java

public static double code(double x) {
	return (x * x) * 0.5;
}

Python

def code(x):
	return (x * x) * 0.5

Julia

function code(x)
	return Float64(Float64(x * x) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * 0.5;
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{2}} \]

   3. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{2} \]

   4. lower-*.f64 99.6

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot 0.5 \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.5} \]
6. Add Preprocessing

Alternative 7: 52.2% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 1.0))

C

double code(double x) {
	return 1.0 - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - 1.0d0
end function

Java

public static double code(double x) {
	return 1.0 - 1.0;
}

Python

def code(x):
	return 1.0 - 1.0

Julia

function code(x)
	return Float64(1.0 - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - 1.0;
end

Wolfram

code[x_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto 1 - \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 52.0%

   \[\leadsto 1 - \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sin x}{1 + \cos x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (sin x) (sin x)) (+ 1.0 (cos x))))

C

double code(double x) {
	return (sin(x) * sin(x)) / (1.0 + cos(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) * sin(x)) / (1.0d0 + cos(x))
end function

Java

public static double code(double x) {
	return (Math.sin(x) * Math.sin(x)) / (1.0 + Math.cos(x));
}

Python

def code(x):
	return (math.sin(x) * math.sin(x)) / (1.0 + math.cos(x))

Julia

function code(x)
	return Float64(Float64(sin(x) * sin(x)) / Float64(1.0 + cos(x)))
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) * sin(x)) / (1.0 + cos(x));
end

Wolfram

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024305 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, A"
  :precision binary64
  :pre (and (<= -0.01 x) (<= x 0.01))

  :alt
  (! :herbie-platform default (/ (* (sin x) (sin x)) (+ 1 (cos x))))

  (- 1.0 (cos x)))